Flow of a Couple Stress Fluid Generated by a Circular Cylinder Subjected To Longitudinal and Torsional Oscillations

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1 Contemporry Engineering Sciences, Vol., 9, no., Flow of Couple Stress Fluid Generted by Circulr Cylinder Subjected To Longitudinl nd Torsionl Oscilltions J. V. Rmn Murthy nd G. Ngrju Deprtment of Mthemtics Ntionl Institute of Technology, Wrngl (A.P), Indi-564. Abstrct The flow generted by circulr cylinder, performing longitudinl nd torsionl oscilltions, in n infinite expnse of couple stress fluid is studied. Anlyticl expressions for the velocity components re obtined under vnishing couple stresses of type A condition or super dherence condition of type B on the boundry. The effects of couple stress prmeter, Reynolds number nd the rtio of couple stress viscosities prmeter on trnsverse nd xil velocity components re presented grphiclly. The drg force cting on the wll of the cylinder is derived nd effects of Couple stress prmeters on drg re shown grphiclly. Keywords: Couple-stress fluid, Longitudinl nd Torsionl Oscilltions, Drg.. Introduction The theory of couple stresses in fluids, developed by Stokes [7] in 966, represents the simplest generliztion of the clssicl theory nd llows for polr effects such s the presence of couple stresses nd body couples. The couple stress theory of fluids defines the rottion field in terms of the velocity field. In fct the rottion vector is equl to one-hlf the curl of the velocity vector s in the cse with

2 45 J. V. Rmn Murthy nd G. Ngrju Newtonin fluids. Second order grdient of the velocity vector, rther thn the kinemticlly independent rottion vector of symmetric hydromechnics, is introduced into the stress constitutive equtions nd consequently the theory yields only one vector eqution to describe the velocity field. The considertion of couple-stress, in ddition to the clssicl Cuchy stress, hs led to the recent development of theories of fluid microcontinu. This new brnch of fluid mechnics hs ttrcted growing interest during pst few decdes minly becuse it possesses the mechnism to describe such rheologiclly complex fluids s liquid crystls, polymeric suspensions nd niml blood for which the Nvier-Stokes theory is indequte. The flow of fluid due to cylindricl rod oscillting with longitudinl nd torsionl motion hs received considerble ttention becuse of its relevnce in mny technicl problems of prcticl importnce such s mixing, oil drilling nd towing opertions. The motion of clssicl viscous fluid due to the rottion of n infinite cylindricl rod immersed in the fluid ws first described by Stokes(886). Externl flows generted due to longitudinl nd torsionl oscilltions of rod were found in the clssicl ppers of Csrell, Lur(969). Rjgopl(983) studied the sme problem for the cse of second grde fluid. The motion of clssicl viscous fluid inside n infinite cylinder ws studied by Rmkissoon(99) nd he derived n nlyticl expression for sher stresses, drg on the cylinder nd velocity ws depicted grphiclly. Cmlet-Eluhu, Mjumdr(998) hve investigted the sme problem for micropolr fluid nd exmined the effect of micropolr prmeters on the two components of the velocity field through grphicl curves by using Mthemtic. Owen nd Rhmn(6) studied the sme type of flow with n Oldroyd-B liquid. Clmelet-Eluhu, Rosenhus() studied, micropolr fluid inside moving infinite circulr cylinder due to its oscilltions long nd bout its xis nd they found nlyticl solutions by pplying lie group methods. Using vrious types of fluids, the flow generted due to longitudinl nd torsionl oscilltions of circulr cylinder ws exmined by few uthors. Rmkissoon et l (99) hve exmined polr fluid by using trnsform methods nd they hve presented the effect of micropolr prmeters on the microrottion nd velocity fields grphiclly. Bndelli et l(994) studied the flow of third grde fluid. Rjgopl nd Bhtngr (995) presented two simple but elegnt solutions for the flow of n Oldroyd-B fluid. In the first prt, they considered the flow pst n infinite porous plte nd found tht the problem dmits n utomticlly decying solution in the cse of suction t the plte nd tht in the cse of blowing it dmits no such solution. In the second prt, they studied longitudinl nd torsionl oscilltions of n infinitely long rod of finite rdius. Pontrelli(997) hs studied the xi-symmetric flow of homogeneous Oldroyd-B fluid with suction or injection velocity pplied t the surfce. Akyildiz(998) studied n Oldroyd-B fluid nd he exmined the effect of the elsticity on the velocity field nd dynmic boundry lyer. Fetcu nd Corin

3 Flow of couple stress fluid 453 Fetcu (6) hve obtined the strting solutions corresponding to the motion of second grde fluid by mens of the finite Hnkel trnsforms. Vieru et l (7) investigted the exct solutions for the motion of Mxwell fluid using Lplce Trnsforms. Krim Rhmn et l(8) studied the motion of viscoelstic incompressible flow of the upper convected Mxwell fluid t different frequencies of oscilltions of the cylinder long nd bout its xis nd he presented velocity components grphiclly for prticulr vlues of the flow prmeters. Mehrdd Mssoudi nd Trn X.Phouc (8) hve solved numericlly the flow of second grde fluid nd they presented the results grphiclly for the sher stresses t the wll. In this pper we consider the flow of couple stress fluid generted by circulr cylinder subjected to longitudinl nd torsionl oscilltions.. Description nd Formtion of the problem: Consider circulr cylinder of rdius within n incompressible iω τ couple stress fluid. The cylinder is subjected to torsionl oscilltions e nd iω τ longitudinl oscilltions e with mplitudes q sinβ, q cos β long the respective directions with ω s the frequency of the longitudinl oscilltions, ω s the frequency of the torsionl oscilltions, q s the mgnitude of oscilltions nd β is the ngle between the direction of torsionl oscilltion nd the bse vector e θ.i.e The cylinder oscilltes with velocity s given by the expression iωτ iω τ Q Γ = q ( Sin β e eθ + Cos β e e z ) nd the flow of the couple stress fluid being generted due to these oscilltions of the cylinder. Choose the cylindricl polr coordinte system(r,θ, Z) with the origin t the center of the cylinder nd Z-xis long the xis of the cylinder. The physicl model illustrting the problem under considertion is shown in figure. Fig Geometry of the problem non dimensionl form After neglecting body forces nd body couples, the field equtions governing the couple stress fluid dynmics s given by Stokes [] re Q = () Q ρ + Q Q = P µ Q η Q () τ

4 454 J. V. Rmn Murthy nd G. Ngrju where Q is velocity, P is pressure, ρ is density, τ is time, µ is viscosity nd η is couple stress viscosity. By nture of the geometry nd flow, the velocity components re ssumed to be xilly symmetric nd depend only on rdil distnce nd time. Hence the velocity is tken in the form Q = V ( R, τ) e θ + W ( R, τ) e z (3) By introducing the following non-dimensionl scheme Q P q R Z q =, p =, t = τ, r = nd z = (4) q ρ q The equtions in (), () nd (3), for the flow tke the following non-dimensionl form. q = (5) q Re + q. q = Re p q S q (6) t ρq η where Re = = Reynolds number nd S = = couple stress prmeter. µ µ Let us choose the velocity vector q nd pressure p in the form q i t i t () r e σ e θ + w () r e σ e z iσ t () e = v (7) p = p r (8) Substituting (7) in (6), nd compring the coefficients of e r, e θ, e z we get dp v = dr r (9) 4 iσ Rev = D v S D v () iv iσ = + + w + w Rew w w S w w 3 r r r r () where D v = v + v v nd D 4 v = D (D v) r r We express the eqution () s 4 iσre D D + v = S S () This cn be written s ( D λ )( D λ ) v = (3) iσre where λ + λ = nd λ λ = S S Similrly the eqution () reduces to 4 iσ Re D D + w = S S (4)

5 Flow of couple stress fluid 455 which cn be written s ( D α )( D α ) w = (5) where iσ Re α + α = nd α α = S S Now the equtions (3) nd (5) re solved for v nd w under the no slip condition nd type A condition or type B condition on the boundry. These conditions re given s follows. No slip Condition The velocity of the fluid on the boundry is equl to the velocity of the boundry. It is iωτ iωτ explicitly given by Q = Velocity of Γ = q ( e e e ) Γ Sin β θ + Cosβ e z It tkes the following in non-dimensionl form iσ t iσ t q = cos β e e θ sin β e e r + = z This condition cn be explicitly written s in the following equtions v () = cosβ nd w ( ) = sinβ (6) Type A Condition Type A-condition represents vnishing of couple stress tensor on the boundry. The constitutive eqution for couple stress tensor M is given by M = mi + η ( Q) + η [ ( Q) ] T (7) i σ v t Tking q = w e e θ + v + e z r in the eqution for M bove we get the expression for M s M ηq η q i σ t ηq η q = m e w + w e + w w rer + eθ eθ + ez ez + er e θ eθer e r r ηq iσ t η q iσt + D v e erez + D v e eze r From this, we get the conditions tht D η v = nd w e w = where e = on r = r η (8) Type B Condition Type B-condition is the super dherence condition on the boundry. This condition requires tht ngulr velocity on the boundry = ω Γ = curlq Γ, which implies tht v w = σ nd v + = on r = r (9) The fluid is t rest s r the solutions of (3) nd (5) cn be written s v = K( λr ) + K( λr) () w = 3 4 K ( αr) + K ( α r) α α () i σt ( )

6 456 J. V. Rmn Murthy nd G. Ngrju Now the constnts,, 3 nd 4 cn be found out numericlly for different vlues of couple stress prmeters by using the boundry conditions (6),(8) nd (9) in () nd (). 3. Clcultion for Drg The drg D cting on cylinder of length L is given by π ( T cos β + T sin β ) d θ D = L 3 () The stress component in () is defined by the following constitutive eqution for couple stress fluids (Stokes []). T T ij = pi + λ( Q) I + µ ( Q+ ( Q) ) + I ( M ) (3) The stress components re clculted from (3) s µq ηq i σt T w w w w e 3 3 = + + nd r r µq v ηq i σt T = v - + v v v v e r r r r The Non-dimensionl form of stress components T 3 nd T on the cylinder cn be clculted s µ q i σ t T3 = [ 3( + Sα ) K( α) + 4( + Sα ) K( α )] e (4) µ q i σ t T = [ ( λ( + Sλ ) K( λ) + SλK ( λ) ) + ( λ( + Sλ ) K( λ) + Sλ K( λ) )] e (5) Now finlly the non dimensionl drg D' is given by D' = T cosβ + T sin on r = (6) ( ) 3 β D where D = πlµ q 4. Numericl clcultion nd Results The nlyticl expression for the non-dimensionl velocity components v, w nd drg re given by the equtions (), () nd (6) respectively. For different vlues of prmeters like couple stress prmeter S, Reynolds number Re nd the rtio of couple stress viscosities prmeter e on velocity components v nd w re computed numericlly nd results re grphiclly presented through figs. In figs 5, the velocity vritions t different times re shown. The drg is clculted numericlly t different times for fixed σ nd σ nd the results re presented through grphs 6 7. When the ngle β = the problem reduces to rotry oscilltions bout the xis of the cylinder. When β =π/, the problem becomes the specil cse of longitudinl oscilltions long the xis of the cylinder. When σ = σ nd oscilltions re periodic, our results re similr to the results of Clmelet-Eluhu nd Mzumdr [7].

7 Flow of couple stress fluid 457 The numericl results for velocities nd drg re computed t S=, σ =.5, e=.5, σ =.5, β =.7, t = π, Re=.7 by fixing one of these vlues in the grphs. The figures for type A boundry condition re shown on left column nd the figures for type B conditions re on the right column. The grphs,3,6 nd 7 indicte tht the condition type A nd the condition type B show no much difference for trnsverse velocity v. But from figures 4, 5, 8 nd 9, in type B condition, the xil velocity w tkes more negtive vles i.e. fluid prticles re pushed downwrds ner to the cylinder. It cn be seen from Figs 5 tht s the couple stress prmeter S increses; the vlues of trnsverse velocity v increse for type A nd type B conditions wheres the vlues of xil velocity w increse for type A condition, but the vlues of w decrese for type B condition. We observe tht from Figs 6 9 tht s the Reynolds number Re increses; the vlues of trnsverse velocity v decrese for type A nd type B conditions wheres the vlues of xil velocity w decrese for type A condition, but these vlues of w increses for type B condition. Type B boundry condition does not involve the prmeter e which is the rtio of couple stress viscosity coefficients η nd η. In type A condition, s it cn be seen from Figs the prmeter e increses the xil velocity w increses, but the effect of e on trnsverse velocity v is not very significnt. i.e., the vrition in the vlues of e does not result in much vrition in the vlues of v. i.e. there is no significnt effect on v due to the prmeter e. The figs 5 indicte the cse of viscous fluids for the velocities v nd w t different times nd our results re in good greement with the observtions of Rmkissoon [6]. The non-dimensionl drg is clculted numericlly for different vlues of non-dimensionl time in multiples of π/σ t fixed vlues of σ, σ nd the results re shown in the Figs 6 7 nd it is observed tht s S increses, the mplitude of oscilltion for the drg lso increses for both type A condition nd type B conditions. i.e., the mplitude of oscilltions for drg in the cse of viscous fluids is less thn tht of the couple stress fluids. Hence the couple stress fluids offer more drg thn tht of viscous fluids. We observe, from fig 8, tht s the prmeter e increses drg increses. Fig Type A vrition of v with r Fig 3 Type B vrition of v with r

8 458 J. V. Rmn Murthy nd G. Ngrju Fig 4 Type A vrition of w with r Fig 5 Type B vrition of w with r Fig 6 Type A vrition of v with r Fig 7 Type B vrition of v with r Fig 8 Type A vrition of w with r Fig 9 Type B vrition of w with r

9 Flow of couple stress fluid 459 Fig Type A vrition of v with r Fig Type A vrition of w with r Fig Type A vrition of vexp(iσ t) Fig 3 Type B vrition of vexp(iσ t) with r with r Fig 4 Type A vrition of wexp(iσ t) Fig 5 Type B vrition of wexp(iσ t) with r with r

10 46 J. V. Rmn Murthy nd G. Ngrju Fig 6 Type A vrition of D' with σ t Fig 7 Type B vrition of D' with σ t Fig 8 Type A vrition of D' with S References [] Stokes V. K. (966): Couple stresses in fluids, Phys. Fluids, Vol 9, pp [] Rmkissoon H. (978): Drg in couple Stress Fluids, ZAMP, Vol 9, pp [3] Stokes G.G. (886): On the effect of rottion of cylinders nd spheres bout their own xes in incresing the logrithmic decrement of the rc of vibrtion.(mthemticl nd Philosophicl Pper-5) Cmbridge: Cmbridge University Press, Englnd, pp 7 4. [4] Csrell M. J nd Lur P.A (969): Drg on oscillting rod with longitudinl nd torsionl motion, J. Hydronut., Vol 3, pp [5] Rjgopl K.R (983): Longitudinl nd Torsionl oscilltion of rod in non-newtonin fluid, Act. Mech., Vol 49, pp 8 85.

11 Flow of couple stress fluid 46 [6] Rmkissoon H nd Mjumdr S.R (99): Flow due to the longitudinl nd torsinl oscilltions of cylinder, ZAMP, Vol 4, pp [7] Clmelet-Eluhu C nd Mzumdr D R (998): Flow of Micropolr fluid through circulr cylinder subject to longitudinl nd torsinl oscilltions. Mthl. Comput. Modelling, Vol 7, pp [8] Owen.D nd Rhmn K (6): On the flow of n Oldroyd-B liquid through stright circulr tube performing longitudinl nd torsinl oscilltions of different frequencies, Mthemtic, Vol 4, pp 9. [9] Clmelet-Eluhu C nd Rosenhus V (): Symmetries nd solutions of micropolr through cylinder. Act. Mech., Vol 47, pp [] Rmkissoon H, Eswrn. C.V nd Mjumdr S.R (99): Longitudinl nd Torsionl oscilltion of rod in polr fluid, Int. J. Engng. Sci, Vol 9(), pp 5. [] Bndelli R, Lpczyk I nd Li H (994): Longitudinl nd torsionl oscilltions of rod in third grde fluid, Int. J. Non-liner Mech, Vol 9, pp [] Rjgopl K. R nd Bhtngr R. K (995): Exct solutions for some simple flows of n Oldroyd B fluid, Act. Mech, Vol 3, pp [3] Pontrelli, G (997): Longitudinl nd torsionl oscilltions of rod in n Oldroyd-B fluid with suction or injection, Act Mech., Vol 3, pp [4] Akyildiz.F.T (998): Longitudinl nd torsinl oscilltions of rod in viscoelstic fluid, Rheol. Act, Vol 37, pp [5] Fetecu. C nd Corin Fetecu (6): Strting solutions for the motion of second grde fluid due to longitudinl nd torsionl oscilltions of circulr cylinder, Int. J. Engng. Sci, Vol 44, pp [6] Vieru D, Akhtr W, Fetecu C nd Corin Fetecu (7): Strting solutions for the oscillting motions of Mxwell fluid in cylindricl domin, Meccnic, Vol 4, pp [7] Lur Bridgll nd Krim Rhmn (8): Upper convected Mxwell fluid due to solid rod oscillting with different frequencies. J. Engng. & Appl. Sci, Vol 3, pp [8] Mehrdd Mssoudi nd Trn X.Phouc (8): On the motion of second grde fluid due to longitudinl nd torsinl oscilltions of cylinder-a numericl study. Appl. Mth. Comput. Vol 3, pp Received: Jnury, 9